No, I don't agree there are " too many endomorphisms" in Set. What makes maths interesting is all those "weird" maps and unexpected things. We're not in the business of outlawing anything that we don't like at first, but come to grips with it.

Another example is where you map the zero-dimensional Cantor set onto the one-dimensional interval, by mapping ternary expansion $\sum 2a_k3^{-k}$ into binary expansion $\sum a_k 2^{-k}$. With a differentiable map (more generally a Lipschitz map) the image has dimension ${}\le$ the domain. But not for non-differentiable maps like Peano. Reportedly, Cantor was astounded when he found there was a bijection between $\mathbb R$ and $\mathbb R^2$. If you think there are too many maps in Set, maybe you should go to a category of differentiable maps or something.

This question was motivated by the example of the grothendieck group: If you try to define the grothendieck group of the category $C$ of countably infinite rank projective $R$-modules, denoted $K_0(C)$, this group is trivial by the Eilenberg trick. How do we define a grothendieck group of $C$ that is "non-trivial"?

@hm2020 the standard trick is to find a reasonable small subcategory of $C$. The philosophy behind Eilenberg's swindle is that something too big has to be homotopically trivial, this "explains" in a sense why $K_0(C)$ should be trivial.

@LeoAlonso - Let $X:=Spec(A)$ is a smooth affine scheme over a field $k$ of characteristic zero. You want the class $[D_X]$ of the ring of differential operators $D_X$ to be non-trivial in $K_0(C)$ - which category $C$ do you choose in this case?

I would choose $D^b_{coh}(\mathcal{D}_X)$, but unfortunately I do not know very much about it. In any case its relation with coherent $\mathcal{O}_X$-sheaves on one side and with constructible $\mathbb{C}_X$-sheaves on the other might give a starter point.

Beauty is in the eye of the beholder... What you find weird I can find beautiful or surprising. I'm more of a Platonist.

@HennoBrandsma - The "weird" maps you speak about makes maths "ugly". Someone at some point said: "Beauty is the first test. There is no permanent place in the world for ugly mathematics" (this was stated by an analyst). I agree to this.

@HennoBrandsma - "We're not in the business of outlawing anything that we don't like at first, but come to grips with it." When you speak of "we" - who are you speaking about?

Mathematicians in general, maybe scientists even.

@HennoBrandsma - so if someone found an alternative approach to set theory where the dimension formula was true for continuous manifolds - you (and others) would not accept this?

@hm2020: that remark about beauty was made by G H Hardy. Anyway your post is a bit long and asks multiple questions. It would be great if you can write multiple posts each asking a single question. If they are related maybe you can give references to other questions.

"If you are out to describe the truth, leave elegance to the tailor" - Einstein. I think Hardy was old and embittered when he wrote that quote about ugly mathematics. His "Mathematician's Apology" is very interesting, but it says more about Hardy than about mathematicians in general.

Math SE is a good place to ask narrow, focused questions. It is not a good place to write long expositions including multiple questions. When you post a question on Math SE, the fundamental content of the question should fit into two or three sentences (though as much supporting detail as you like can also be included). If you have multiple questions, you should separate them into separate posts.

No, I don’t think so. And Peano curves aren’t continuous manifolds..

I have voted to close as "needs more focus". If the question were less of a rant, I might have written an answer about o-minimality, which has been advertised by some as a partial realization of Grothendieck's idea of "tame topology". The idea here is not to change our foundations, but to restrict our attention to sets and functions which are definable in some restricted first-order theory, e.g. the real numbers with certain elementary functions. Assuming o-minimality of the theory, we obtain a good dimension theory and other regularity properties for definable sets and functions.

The problem with "infinity" arise in algebra, geometry, analysis, topology - I wrote the questions in one post to illustrate this. Maybe an "amateur" reads this thread and solves the problem. It is an "elementary" question and you do not need to know a lot about set theory to understand the problem. If you do this - please post the solution on this site.

Also, your idea of outlawing injective but not surjective functions is definitely not sound. In Example 1, you talk about "another copy of Z". Keeping track of multiple copies of structures all the time sounds much uglier to me than Peano curves. But even if you manage to make this precise in an elegant way, any two copies of Z will have a canonical bijection between them (otherwise I can't imagine what's meant by "copy"). Composing the multiplication by 2 map (from Z to its copy) with this canonical bijection (from the copy back to Z), we get the multiplication by 2 map from Z to itself.

So if you want to disallow injective but not surjective functions, you either need to give up (1) simple arithmetic on Z and N, or (2) composition of arbitrary functions, or (3) countably infinite sets as real mathematical objects. None of which is attractive if you want to do any mathematics...

@AlexKruckman - my point is the following: The "Peano curve" is so counterintuitive to me that my personal feeling is that ZF set theory is inconsistent. For a finite set $S$ an endomorphism $f$ is injective iff it is surjective. A "naive" first attempt was to make this assumption for arbitrary sets: Given an arbitrary set $S$ and an endomorphism $f \in Hom(S,S)$. We postulate that $f$ is injective iff $f$ is surjective. This is a condition on $Hom(S,S)$ - hence we make a restriction on the Hom sets in our category $Sets$. One question is: Does this lead to an interesting category $Sets$?

@HennoBrandsma - "And Peano curves aren’t continuous manifolds." The "Peano curve" is by definition the image of the map $f(t)$ defined in my post. The sets $I,X$ are by definition continuous manifolds with corners. The map $f$ is a continuous map.

@hm2020 The point of my previous comment is: it doesn't even make a category, since you can't compose in general. If you feel ZF set theory is inconsistent, you are welcome to spend your time trying to prove a contradiction from ZF - but note that a far weaker set theory than ZF can still define the Peano curve, and any contradiction in this weaker system will likely take a large chunk of "ordinary real analysis" with it.

Regarding your most recent edit: in o-minimality, we can define a category of "definable manifolds" (all of which will be finite dimensional) and definable maps, and the Proposition on dimensions at the end of your question is a theorem. Check out the book "tame topology and o-minimal structures" by Lou van den Dries.

@AlexKruckman - "If the question were less of a rant, I might have written an answer about o-minimality, which has been advertised by some as a partial realization of Grothendieck's idea of "tame topology"." My question was an honest question on ZF set theory motivated by some paradoxes - I do not agree to the claim that the post was "rant". If this site is to be friendly to "amateurs" some of the users should change their attitude.

@hm2020 This has nothing to do with "amateurs" vs "professionals". Your post is long and digressive. You go off in a couple of different directions, and ask (seemingly) multiple questions. "What are the properties of $b$?" is not a good question---it is too broad and unfocused. You need to come up with an "elevator pitch" for this question: in two or three sentences, with no other context, explain the question. This explanation should not rely on several pages of examples and explanation---in two sentences, what is it that you want to know?

@XanderHenderson - "Is there a way to change the definition of the "category of sets" such that the dimension formula holds for the category of continuous manifolds (with corners) and surjective continuous maps?" - This is a very specific question in my opinion. I ask for an alternative definition of the category of sets. The reason my post is long and involved is because problems and paradoxes related to "infinity" and infinite sets arise in all parts of mathematics as I indicate above: In topology,algebra, homological algebra, analysis, number theory etc.

@XanderHenderson - this is why I ask you to "unblock" the post and let "amateurs" all around the world try to answer the questions I pose.

@hm2020 You have not done any of the things which I have asked you to do. Your post is still long and digressive, and it is difficult to discern what the actual question is from your post. That you can summarize the question in a comment is great, but not everyone is going to read the comments. Please edit your question to narrow it down and focus on this one question that you are interested in.

Try to trim out the tangential points and the bits that don't help the reader to understand the core of what you are after.